In the Feynman lectures on physics, Feynman gives an example where the Heisenberg uncertainty relation must hold true (http://www.feynmanlectures.caltech.edu/III_01.html). Specifically he writes: "Let us show for one particular case that the kind of relation given by Heisenberg must be true in order to keep from getting into trouble. We imagine..." (section 1.8 near the bottom of the page).

However, I am not sure I follow his argument, because in his argument, he invokes the uncertainty principle itself on the ability to measure the momentum of the wall. Specifically, he writes (in explaining why the proposed way of measuring the momentum of the electron won't work) "So when we measure the momentum after the electron goes by, we can figure out how much the plate’s momentum has changed. But remember, according to the uncertainty principle we cannot at the same time know the position...".

This seems to be a circular argument. The purpose of the example was to
show why the uncertainty relation must hold in this situation. But in the argument, he invokes the uncertainty relation. And actually the argument does not in any way produce formulae or otherwise showing the uncertainty relation itself, it all rests on invoking the uncertainty relation itself. So it seems you could have substituted in almost any claim for an alternative uncertainty relation and then "proven" that this must hold in this situation by using this argument as a template, invoking the claimed alternative uncertainty principle when you come to the determination of the movement of the plate.

2 Answers
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"Let us show for one particular case that the kind of relation given by Heisenberg must be true in order to keep from getting into trouble. We imagine..." This is a slightly informal statement, in Feynman's flowing style. The background is something like, 'we have argued that when the electrons go through one slit at a time, no fringes are seen, but when both slits are open, we get fringes. But surely we could examine the electrons as they go. If we see them pass through one slit or the other, this amounts to the first case (one slit at a time), but now we have both open slits, so the whole subject is in danger of being inconsistent. What happens then?'

Now we come to Feynman's statement as quoted. He is saying that some sort of general principle is going to have to be true, if the subject is going to be able to make sense ('in order to keep from getting into trouble'), a principle or relation which has the effect that a physical process that can determine which slit the electron passed through must result in loss of interference fringes. So he is indeed assuming Heisenberg's uncertainty regarding position and momentum, and his aim is to show that it is sufficient to 'do the work' of keeping the subject consistent. To be precise, he is showing this only in one illustrative case. The aim is to show what happens in this case, and thus to help the student get a feel for how the physics pans out more generally.

In short, at this point Feynman is assuming the correctness of quantum theory, and expounding its physical implications. He is especially interested in expounding its internal consistency and robustness against possible objections.

Here I devise an experiment where I can measure the momentum of electrons (by observing the plate), and know which hole they went through.

Doing this does not perturb the electrons, so the pattern should form.

However, it will not form and a numerical computation (announced in the text but not carried out) will prove that the holes move just enough to prevent the pattern from forming.

This thought experiment is a smart way of transposing quantum measures (on the electrons) to macroscopic measures (on the plate). And it "proves" that because the inequalities hold, this smart way just does not suffice, and you cannot trick quantum mechanics into forming the pattern while determining which slit the electrons went through...